On the Robustness of Ljung−Box and McLeod−Li Q Tests:
A Simulation Study
Yi−Ting Chen
Sun Yat−Sen Institute for Social Sciences and Philosphy, Academia Sinica
Abstract
In financial time series analysis, serial correlations and the volatility clustering effect of asset returns are commonly checked by Ljung−Box and McLeod−Li Q tests and filtered by ARMA−GARCH models. However, this simulation study shows that both the size and power performance of these two tests are not robust to heavily tailed data. Further, these Q tests may reject processes without ARMA−GARCH structures simply because of nonlinearity and conditionally heteroskedastic higher−order moments. These results imply that, to avoid misleading interpretations on time series data, these two tests should be used with care in practical applications.
Citation: Chen, Yi−Ting, (2002) "On the Robustness of Ljung−Box and McLeod−Li Q Tests: A Simulation Study." Economics
1
Introduction
Autocorrelation is a leading measure of serial dependence in time series analysis. To understand the dependence structure of time series data, researchers routinely calculate sample autocorrela-tions and check the joint significance of these statistics by the Q test of Box and Pierce (1970, theQBP test) or Ljung and Box (1978, the QLB test). In financial time series analysis, it is par-ticularly important to check serial correlations of squared series; such dependence is referred to as the volatility clustering effect. TheQ test of McLeod and Li (1983, the QML test) is used for this purpose. TheQML test is indeed a particular QLB test based on sample autocorrelations of squared series, and it is asymptotically equivalent to the LM test of Engle (1982). To explain the serial correlations and volatility clustering effect detected by theseQ tests, empirical researchers often filter time series data by certain autoregressive moving averages and generalized autoregres-sive conditional heteroskedasticity (ARMA-GARCH) models. TheseQ tests are also applied on the residuals of estimated models as diagnostic checks. Although this modelling strategy is quite popular in empirical studies, it may encounter some problems in practice.
Note that the above Q tests require that the process being tested be an independently and identically distributed (i.i.d.) sequence. This condition is much more stringent than the hypothe-sis of white noise, so that theQBP andQLB tests (theQML test) may reject the null hypothesis not only for serial correlations (the volatility clustering effect) but also for other types of serial depen-dence. Further, theQBP andQLB tests (theQML test) need data to possess finite fourth (eighth) moment. Therefore, the performance of these tests is likely to be influenced by the fatness of distributional tails. In fact, most risky assets’ returns have heavily tailed distributions. Several studies even reported that such distributions may not have finite fourth moment; see Jansen and de Vries (1991), Loretan (1994) and Phillips, and de Lima (1997). Consequently, it is important to investigate the robustness of theseQ tests in these situations.
For this purpose, we conduct a Monte Carlo simulation to assess finite sample performance of the QLB and QML tests. The QBP test will not be discussed in this study because the QLB test is already a finite-sample-correctedQBP test. The simulation results show that distributional heavy-tails may distort the asymptotic null distributions of the QLB and QML tests and reduce the power of these tests. Further, these Q tests may reject their null hypotheses because of nonlinearity or conditionally heteroskedastic skewness, even if the process being tested does not have any ARMA-GARCH structures. In consequence, theseQ tests should be used with care to avoid yielding misleading interpretations on time series data, and it is important to consider the use of tests for moment conditions, nonlinearity, and independence to complement theseQ tests before accepting ARMA-GARCH models.
The rest of this paper is organized as follows. In Section 2, we write the notation and the QLB and QML test statistics. In Section 3, we introduce the simulation experiment design. The
2
The Ljung-Box and McLeod-Li
Q tests
Let{yt} be a time series with the lag-k autocorrelation: ρ1(k) =
cov(yt, yt−k)
var(yt) , k = 1, 2, . . . . The lag-k sample autocorrelation of {yt} is
ˆ ρ1(k) = T t=k+1(yt− ¯yT)(yt−k− ¯yT) T t=1(yt− ¯yT)2 ,
whereT denotes the sample size; ¯yT is the sample mean of {yt}. To test if {yt} is a white noise sequence, it is common to check the hypotheses consisted of the firstm autocorrelations:
Ho : ρ1(1) =ρ1(2) =. . . = ρ1(m) = 0,
H1 : ρ1(k) = 0, for somek = 1, 2, . . . , m.
For these hypotheses, theQLB test statistic is of the form: QLB(m) = T (T + 2) m k=1 ˆ ρ2 1(k) T − k.
This statistic has the asymptotic null distributionχ2(m) provided that {y
t} is an i.i.d sequence
and thatyt has a finite fourth moment.
The QML test is a particularQLB test based on sample autocorrelations of the squared series {y2
t}. Its test statistic is
QML(m) = T (T + 2) m k=1 ˆ ρ2 2(k) T − k, where ˆ ρ2(k) = T t=k+1(y2t − ˆσ2T)(yt−k2 − ˆσ2T) T t=1(y2t − ˆσ2T)2 , σˆT2 = 1 T T t=1 y2 t.
This statistic also has the asymptotic null distribution χ2(m) provided that {yt} is an i.i.d. sequence and that yt possesses a finite eighth moment. This test is often used to check the whiteness of{y2 t}: ρ2(k) := cov(yt2, yt−k2 ) var(y2 t) = 0, for all k = 1, 2, . . . , m,
against the volatility clustering effect. As we have noted previously, the QLB and QML tests also serve as diagnostic checks for estimated ARMA-GARCH models. However, in this case the degrees of freedom of the asymptotic null distribution ofQLB(m) and QML(m) have to be reduced by the number of parameters estimated.
3
The data generation processes
In this simulation experiment, we consider five types of data generation processes (DGPs): i.i.d. sequences, a simple nonlinear process, a conditionally heteroskedastic skewness process, an au-toregressive (AR) process, and a GARCH process with different innovation distributions. The first three DGPs are designed to represent processes without ARMA-GARCH structures. Sup-posedly, theQLB and QML tests should be powerful against the last two DGPs but not the first three DGPs. The details of these DGPs are described as follows.
IID sequences
In this case,
yt=εt, (1)
where {εt} is an i.i.d. sequence. To study the influence of the tail behavior of εt on the Q tests, we consider three distributions for εt: the standard normal distribution N(0, 1), Student t distribution with the degrees of freedom ν, denoted as t(ν), and the standardized log-normal distribution:
εt= exp(λut− λ
2/2) − 1
(exp(λ2)− 1)1/2 , ut∼ N(0, 1),
with the skewness parameterλ, denoted by L(λ). The latter two distributions can yield different tail properties by adjusting their parameters. It is well known thatt(ν) is a symmetric distribution without finiteν-th moment. As compared to N(0, 1), t(ν) has heavier tails and higher peakedness when ν is small. As ν → ∞, this distribution converges to N(0, 1). On the other hand, L(λ) is a right-skewed distribution with zero mean and unit variance. The fatness of its right tail, and hence skewness and kurtosis, increase with the value of |λ|. This distribution also encompasses N(0, 1) as λ → 0; see Johnson et al. (1995).
A simple nonlinear process
This nonlinear process is of the form: yt=
µ1+σ1εt, yt−1> 0,
µ2+σ2εt, yt−1≤ 0,
(2)
where {εt} is an i.i.d. N(0, 1) sequence. The magnitudes of |µ1 − µ2| and |σ1− σ2| control the
significance of the location and scale switching behavior of {yt}, respectively. Obviously, this process has no ARMA-GARCH structures. Granger and Ter¨asvirta (1999) used a particular case of this process, (µ1, µ2;σ1, σ2) = (1, −1; 1, 1), to illustrate that autocorrelations may yield
misleading linear properties.
Some studies recently aimed at investigating conditionally heteroskedastic skewness of asset returns; see e.g., Hansen (1994) and Harvey and Siddique (1999). To represent this DGP, we consider that yt is distributed as L(λt) with the skewness parameter λt, which follows the first-order AR process:
λt=δo+δ1λt−1+εt, (3)
where {εt} is an i.i.d. N(0, 1) sequence. The process {yt} has zero conditional mean and unit conditional variance, so that it does not have serial correlations and the volatility clustering effect. However, it is still a serially dependent process governed by the conditionally heteroskedastic skewness parameters{λt}.
An AR process
This case considers the first-order AR process:
yt=αyt−1+εt, (4)
where{εt} is an i.i.d. sequence with N(0, 1), t(ν), or L(λ). In this DGP, {yt} has serial correla-tions but no volatility clustering effect.
A GARCH process
This GARCH process is of the form:
yt=εth1t/2, ht=βo+β1y2t−1+β2ht−1, (5)
where{εt} is an i.i.d. sequence with N(0, 1), t(ν), or L(λ). This process has volatility clustering effect but no serial correlations.
4
The simulation results
To implement this experiment, we considerT = 100, 500, m/T = 5%, 10%, 30%, ν = 1, 3, 5, λ = 0.5, 1, 2, α = ±0.3, ±0.7, (βo, β1, β2) = (1, 0.3, 0.6), (1, 0.05, 0.9), (µ1, µ2;σ1, σ2) = (1, −1; 1, 1),
(1,1;1,4), (1,-1;1,4), (δo, δ1)= (1,0.3),(1,0.5),(1,0.7),(1,0.9), and many other parameter settings.
However, due to the reason of space limitations, we only report the results of T = 500 with selected parameter settings. Other simulation results are available upon request. For comparing χ2(m) and the empirical distributions of Q
LB(m) and QML(m), the nominal level θ considered
includesθ =50%, 25%, 10%, 5%, and 1%. In Table 1, we show the empirical rejection frequencies of theQLB andQML tests for DGPs (1), (2), and (3). The results of DGPs (4) and (5) are shown in Table 2. These rejection frequencies are calculated from 5000 replications.
4.1 DGPs without ARMA-GARCH structures
We first discuss DGP (1), the case of i.i.d. sequences. For the distributionN(0, 1), Table 1 shows that the empirical rejection frequencies ofQLB(m) and QML(m) are quite close to their respective nominal levels whenm/T = 5%, 10%. This result supports that χ2(m) is a proper approximation to the null distributions of these two statistics in this standard case. However, whenm/T = 30%, the empirical sizes ofQLB(m) and QML(m) are 9.6% and 8.3% (4.0% and 3.2%) for θ = 5% (1%), respectively; that is, theseQ tests are somewhat over-sized for large m/T . This evidence is likely caused by the fact that the variance ofQLB(m) is larger than which of χ2(m) when m/T is large; see Davies et al. (1977).
de Lima (1997) reported that theQMLtest is sensitive to the heavy tails of Pareto distributions. Table 1 further shows that both the QLB and QML tests are not robust to the tail behavior of t(ν) and L(λ). For example, given the distribution L(2) and m/T = 5%, the empirical rejection frequencies of the QLB (QML) test are 20.6%, 14.5%, 10.5%, 8.6%, 6.6% (10.8%, 9.1%, 7.6%, 6.9%, 5.6%) for θ =50%, 25%, 10%, 5%, and 1%, respectively. This illustrates that χ2(m) is inappropriate to approximate the null distributions ofQLB(m) and QML(m) in this case. Similar examples can also be found for other distributions listed in Table 1. From this table, we also observe that theQML test is much more sensitive to heavy tails of data. This is because theQML test requires a stricter moment condition than theQLB test.
Table 1 also shows that the QLB and QML tests are not only sensitive to the tail behavior of data but also not robust to nonlinearity and conditionally heteroskedastic higher-order moments of the process being tested. Recall that DGP (2) with (µ1, µ2;σ1, σ2) = (1, −1; 1, 1) is a nonlinear
process with switching locations and a fixed scale parameter. In this case, the QLB test rejects the null with the frequency 100% for all values ofθ and m/T considered. On the other hand, the QML test has proper size performance when m/T = 5%, 10%. This result reminds us that this
nonlinear process might be fitted by a misleading ARMA model, if the significance ofQLB(m) is misleadingly interpreted as a signal of the need for certain ARMA models to filter the “detected serial correlations” of data. In fact, theQML test may also over-reject the null hypothesis when this nonlinear process has regime switching scale parameters. For example, given (µ1, µ2;σ1, σ2) =
(1, −1; 1, 4), θ = 5%, and m/T = 5%, the empirical rejection frequency of the QML test is 29.4%; even though, this process has no GARCH structures.
For the conditionally heteroskedastic skewness process with (δo, δ1) = (1, 0.5), given m/T =
5%, the QLB (QML) test has the empirical rejection frequencies 36.1% and 3.4% (17.9% and 6.6%) for θ =50% and 1%, respectively. Hence, χ2(m) is not a proper approximation to the null distributions of QLB(m) and QML(m) in this case. For certain parameter settings, the QLB and QML tests may reject their null hypotheses with high probabilities, even if there is no serial correlations and the volatility clustering effect. For instance, given (δo, δ1) = (1, 0.9) and m/T =
4.2 DGPs with an AR or GARCH structure
For the AR (GARCH) process, as expected, Table 2 shows that theQLB (QML) test can powerfully reject the null hypotheses when the innovations are distributed as N(0, 1). For example, given m/T = 5% and θ = 5%, the QLB test has the power 99.4% against the AR process with α = 0.3;
theQMLtest rejects the GARCH process with (βo, β1, β2) = (1, 0.3, 0.6) and (1, 0.05, 0.9) with the
frequencies 99.3% and 87.0%, respectively. However, the power performance of these two tests is still influenced by the tail behavior of innovation distribution. Given m/T = 10% and θ = 5%, the power of theQLB test against the AR process withα = 0.3 drops from 96.8% to 70.8% when the innovation distributionN(0, 1) is replaced by L(2); on the other hand, the QML test has the power 98.3% against the GARCH process with (βo, β1, β2) = (1, 0.3, 0.6) when the innovations
are distributed asN(0, 1), but the power is reduced to 36.2% for L(1) distributed innovations. From this study, we also see that the QLB (QML) test may reject the null hypothesis with certain probabilities because of the volatility clustering effect (serial correlations). Table 2 shows that, given m/T = 5% and θ = 5%, the QLB test has the rejection frequency 85.9% for the GARCH process with (βo, β1, β2) = (1, 0.3, 0.6) and N(0, 1), even though this GARCH process
is a white noise sequence. On the other hand, given the samem/T and θ, the QML test has the rejection frequency 16.5% against the AR process with α = 0.3 and N(0, 1) innovations. This process contains no GARCH structures, but the above rejection frequency is much larger than the 5% nominal level. This evidence demonstrates the difficulty of using these two Q tests to distinguish between serial correlations and the volatility clustering effect.
5
Conclusions
This simulation study shows that both the size and power performance of theQLB andQML tests are not robust to heavily tailed data. Furthermore, these two tests may reject their null hypotheses because of certain types of nonlinearity and conditionally heteroskedastic higher-order moments, even if the process being tested does not have ARMA-GARCH structures. For the processes with AR or GARCH structures, these two tests may not be able to distinguish between serial correlations and the volatility clustering effect. In consequence, we may not completely rely on these Q tests to construct and check ARMA-GARCH models. The results of this study also support the importance of developing new portmanteau tests for serial correlations which can be robust to the volatility clustering effect; see e.g., Lobato et al. (2001) and the references therein. However, we still need other portmanteau tests for serial correlations and the volatility clustering effect that are robust to nonlinearity, conditional heteroskedastic higher-order moments, and distributional heavy tails.
References
Box, G. E. D. and D. Pierce (1970). Distribution of autocorrelations in autoregressive moving average time series models, Journal of the American Statistical Association, 65, 1509–1526. Davies, N., C. M. Triggs, and P. Newbold (1977). Significance of the Box-Pierce portmanteau
statistics in finite samples, Biometrika, 64, 517–522.
de Lima, P. J. F. (1997). On the robustness of nonlinearity tests to moment condition failure, Journal of Econometrics, 76, 251–280.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987–1006.
Granger, C. W. and T. Ter¨asvirta (1999). A simple nonlinear time series model with misleading linear properties, Economics Letters, 62, 161–165.
Hansen, B. E. (1994). Autoregressive conditional density estimation, International Economic Review, 35, 705–730.
Harvey, C. R. and A. Siddique (1999). Autoregressive conditional skewness, Journal of Finance and Quantitative Analysis, 34, 465–487.
Jansen, D. W. and C. G. de Vries (1991). On the frequency of large stock returns: Putting booms and busts into perspective, Review of Economics and Statistics, 73, 18–24.
Johnson, N. L., S. Kotz, and N. Balakrishnan (1995), Continuous Univariate Distributions- Vol-ume 2, New York: Wiley.
Ljung, G. M. and G. E. P. Box (1978). On a measure of lack of fit in time series models, Biometrika,
65, 297–303.
Lobato, I., J. Nankervis, and N. E. Savin (2001). Testing for Autocorrelation Using a Modified Box-Pierce Q Test, International Economic Review, 42, 187–205.
Loretan, M. and P. C. B. Phillips (1994). Testing the covariance stationarity of heavy-tailed time series, Journal of Empirical Finance, 1, 211–248.
McLeod, A. I. and W. K. Li (1983). Diagnostic checking ARMA time series models using squared-residual autocorrelations, Journal of Time Series Analysis, 4, 269–273.
Table 1: Rejection frequencies of QLB(m) and QML(m) for DGPs (1), (2), and (3). Ljung-BoxQ(m) McLeod-LiQ(m) DGP m/T θ = 50% 25% 10% 5% 1% 50% 25% 10% 5% 1% (1) i.i.d. sequences 5% 50.7 24.8 10.4 5.6 1.3 44.9 22.2 9.7 5.4 1.5 N(0, 1) 10% 48.2 25.1 11.4 6.6 1.6 44.9 23.4 10.2 5.8 1.7 30% 48.0 28.5 14.7 9.6 4.0 41.8 24.3 12.3 8.3 3.2 5% 44.2 22.4 9.1 5.0 1.4 20.1 13.6 9.3 7.3 4.8 t(3) 10% 43.0 21.4 9.7 5.7 1.7 20.9 14.2 9.4 7.4 4.3 30% 39.1 22.9 12.1 7.4 2.9 16.9 10.2 5.9 4.2 2.3 5% 16.2 12.5 9.5 7.8 5.8 7.7 6.5 5.5 5.0 4.2 t(1) 10% 16.4 12.4 9.5 8.3 5.9 8.9 7.5 6.3 5.9 4.9 30% 10.7 6.9 4.1 2.9 1.2 4.5 2.6 1.4 0.8 0.4 5% 38.6 19.3 8.8 5.5 2.1 18.3 13.8 10.9 9.3 6.4 L(1) 10% 36.4 19.1 9.3 5.9 2.2 19.1 13.9 10.4 8.4 6.0 30% 34.0 19.6 10.8 7.2 2.9 13.9 8.5 4.9 3.5 1.7 5% 20.6 14.5 10.5 8.6 6.0 10.8 9.1 7.6 6.9 5.6 L(2) 10% 21.9 14.7 10.5 8.0 4.9 12.3 10.4 8.5 7.6 6.0 30% 17.2 10.9 6.6 4.4 1.9 7.7 4.6 2.5 1.8 0.7 (2) nonlinear process with the parameters (µ1, µ2;σ1, σ2)
5% 100 100 100 100 100 47.7 23.9 9.7 5.3 1.5 (1,-1;1,1) 10% 100 100 100 100 100 47.6 24.7 10.8 5.8 1.4 30% 100 100 100 100 100 45.2 26.3 14.2 8.8 3.6 5% 99.5 98.2 95.5 92.7 85.7 70.8 53.3 37.9 29.4 17.5 (1,-1;1,4) 10% 98.6 95.8 91.0 86.6 76.4 61.6 43.9 29.3 22.4 12.7 30% 94.9 88.9 80.9 75.0 61.9 49.9 35.3 24.3 18.7 10.5 (3) conditionally heteroskedastic skewness process with the parameter (δo, δ1)
(1,0.5) 5% 36.1 20.8 11.1 7.5 3.4 17.9 13.9 10.5 9.0 6.6 10% 33.1 18.6 10.3 6.8 3.2 17.0 13.4 10.4 8.9 6.2 30% 28.7 16.7 9.7 6.3 2.8 11.2 7.1 4.5 3.2 1.6 (1,0.9) 5% 62.2 59.9 57.7 56.0 52.9 39.1 36.9 35.2 34.0 32.0 10% 50.4 47.4 44.3 42.7 39.2 31.4 29.8 28.2 27.0 24.9 30% 14.6 12.3 10.9 10.2 8.5 8.9 7.7 6.8 6.3 5.4 Note: The entries are rejection frequencies in percentages;θ denotes the nominal size; T = 500.
Table 2: Rejection frequencies of QLB(m) and QML(m) for DGPs (4) and (5).
Ljung-BoxQ(m) McLeod-LiQ(m) DGP m/T θ = 50% 25% 10% 5% 1% 50% 25% 10% 5% 1% (4) AR(1) process with the parameterα = 0.3 and the distribution of εt
5% 100 100 99.7 99.4 97.5 65.1 42.3 24.1 16.5 7.3 N(0, 1) 10% 99.9 99.6 98.4 96.8 92.0 59.5 37.4 20.8 14.1 5.9 30% 99.3 97.2 93.1 89.3 78.7 53.3 34.5 21.0 15.0 7.4 5% 99.9 99.7 99.4 98.9 96.4 32.2 22.6 15.9 12.5 8.7 t(3) 10% 99.9 99.3 98.1 96.4 89.6 26.8 19.2 13.4 11.0 7.1 30% 97.1 94.0 88.9 84.8 73.7 21.1 14.2 09.2 6.8 3.6 5% 99.5 99.3 98.9 98.5 95.7 8.2 7.2 6.2 5.7 4.8 t(1) 10% 88.4 67.2 56.2 50.1 39.9 9.7 8.3 7.2 6.6 5.6 30% 44.3 38.7 33.9 30.4 24.2 6.1 4.0 2.5 1.8 0.8 5% 100 100 99.9 99.7 97.4 27.4 21.4 16.7 14.7 10.9 L(1) 10% 99.9 99.4 98.2 96.0 87.4 24.7 18.3 14.1 11.5 8.2 30% 95.8 91.9 85.0 79.9 66.9 16.6 11.3 7.0 5.2 2.6 5% 99.9 99.9 99.8 99.8 97.4 13.4 11.2 9.5 8.6 7.1 L(2) 10% 97.2 87.2 77.7 70.8 58.5 14.5 12.5 10.6 9.6 7.8 30% 64.2 57.0 49.1 44.0 34.7 8.7 5.9 3.5 2.7 1.3 (5) GARCH(1,1) process with the parameters (βo, β1, β2) and the distribution ofεt
(1,0.3,0.6) 5% 96.9 93.2 89.2 85.9 78.6 99.8 99.6 99.5 99.3 99.0 N(0, 1) 10% 93.6 87.5 81.1 76.8 68.4 99.1 98.7 98.5 98.3 97.8 30% 69.4 58.9 50.7 45.6 35.9 95.3 94.6 93.7 93.0 91.9 (1,0.05,0.9) 5% 83.4 67.2 51.0 41.3 26.5 95.6 92.7 89.3 87.0 81.8 N(0, 1) 10% 81.4 65.6 50.1 41.4 28.4 93.0 89.1 84.5 81.7 76.6 30% 68.5 52.6 38.8 31.3 19.5 83.4 77.4 71.7 67.7 61.2 (1,0.3,0.6) 5% 99.7 99.3 98.6 97.4 95.3 98.9 98.6 98.4 98.1 97.8 t(3) 10% 98.9 97.8 96.4 95.1 92.3 97.7 97.2 96.7 96.3 95.5 30% 81.3 75.8 69.2 65.1 57.2 90.9 89.7 88.2 87.5 85.9 (1,0.05,0.9) 5% 95.9 90.6 83.1 77.2 66.6 97.8 96.9 95.7 95.2 93.3 t(3) 10% 96.4 91.6 85.6 81.6 71.8 96.5 95.1 94.0 93.3 91.6 30% 85.3 77.8 69.9 64.3 54.0 89.4 87.5 85.4 83.7 80.8 (1,0.3,0.6) 5% 79.5 64.4 49.2 41.7 30.1 62.3 55.1 49.4 45.8 40.4 L(1) 10% 67.6 51.5 38.1 31.3 21.3 50.9 44.3 38.9 36.2 30.6 30% 44.6 30.5 20.7 15.9 8.9 31.1 25.1 19.9 16.9 12.1 (1,0.05,0.9) 5% 74.7 56.9 40.5 32.4 21.0 46.3 39.2 34.3 31.4 26.4 L(1) 10% 68.0 50.0 34.8 27.7 17.8 42.1 35.1 29.5 25.9 21.0 30% 47.0 33.1 21.5 16.1 8.9 22.7 16.6 11.3 8.8 5.8 Note: The entries are rejection frequencies in percentages;θ denotes the nominal size; T = 500.